1. June 25, 2006 Propensity Score Adjustment in Survival Models Carolyn Rutter Group Health Cooperative rutter.c@ghc.org AcademyHealth, Seattle WA

2. June 25, 2006 Outline Propensity Scores: General Ideas Background: depression & mortality among type 2 diabetics Propensity Scores applied to depression & mortality

3. June 25, 2006 Example: Is depression associated with increased mortality in type 2 diabetics? Underlying question: Does depression increase the risk of death ? Estimate the causal effect of treatment on response exposure outcome Z Y AcademyHealth, Seattle WA

4. June 25, 2006 Propensity Scores Propensity score: the probability that a person receives treatment, or is exposed, given a set of observed covariates, X. Randomized Study: P(Tx)=0.5, the propensity score is independent of patient characteristics and the distribution of P(Tx) is the same across treatment groups. Observational Study: P(Tx|X) depends on patient characteristics and differs between treatment groups (because Tx is associated with covariates), so that the treated group has a higher propensity for treatment than the untreated group. AcademyHealth, Seattle WA

5. June 25, 2006 Basic Ideas behind Propensity Score Methods Reduce bias by comparing treated and untreated individuals who have the same propensity for treatment/exposure Key assumption: Strongly Ignorable Treatment Assignment The outcome is conditionally independent of treatment assignment given observed covariates Y P(Z|X) After adjusting for observed covariates, treatment assignment doesnt inform the response. No unmeasured confounders. AcademyHealth, Seattle WA

6. June 25, 2006 Depression & Mortality among Type 2 Diabetics Depression is common in patients with type 2 diabetes 11% to 15% meet criteria for major depression Depressed diabetic patients tend to have –poorer self-management ( diet, exercise, blood glucose checks ) –more lapses in refilling prescribed medications ( oral hypoglycemics, lipid lowering, anti-hypertensive ) –have cardiac risk factors ( smoking, obesity, sedentary lifestyle ) Studies have linked depression to increased mortality among diabetics, but these used a small number of patients, with medical diagnoses based on self report AcademyHealth, Seattle WA

7. June 25, 2006 The Pathways Study: a population-based epidemiologic study of over 4000 patients with diabetes enrolled in an HMO. 4262* included in following analyses 513 with major depression 3749 without major depression Katon, Rutter, Simon et al The association of comorbid depression with mortality in patients with type 2 diabetes. Diabetes Care. 2005 Nov; 28(11):2668-72. AcademyHealth, Seattle WA Depression & Mortality among Type 2 Diabetics

8. June 25, 2006 3 year Mortality Outcome All-cause mortality: May 2001(start recruitment) – May 2004 5/1/2001 – 12/31/2003 (first 31 months): GHC automated health care records + Washington State mortality data 90% of deaths in the State mortality data were in GHC records 1/1/2004 – 4/30/2004 (last 5 months): GHC data alone. Censoring at the end of the study or disenrollment Deaths over a 3-year period: 336 ( 9.0%) in 3749 patients without major depression 60 (11.7%) in 497 patients with major depression

9. June 25, 2006 Proportional Hazards Model Survivor function: S(t) = Pr(T*>t)=1-F(t) T* event time Hazard function: instantaneous event rate Cox proportional hazards model Unspecified Baseline hazard AcademyHealth, Seattle WA

10. June 25, 2006 PH Model Results AcademyHealth, Seattle WA * Known confounders: gender, age, race/ethnicity, education Potential behavioral and disease severity confounders &/or mediators: BMI, current smoker, sedentary lifestyle, HbA1c, use of oral hypoglycemics, use of insulin, complications of diabetes, (pharmacy-based) comorbidity measure (excluding depression meds) MethodEstimateSe(estimate)HRP-value Unadjusted0.340.141.40<0.02 Minimum Adjustment* 0.770.142.16<0.001 Full Adjustment 0.260.161.300.09

11. June 25, 2006 Z Depression X Self Care Disease Severity Age, Sex Education Y Death mediator common cause

12. June 25, 2006 Z X Y mediator common cause

13. June 25, 2006 Propensity Score Adjustment: 3-Step Process 1.Estimate propensity score 2.Evaluate covariate balance given propensity scores 3.Incorporate propensity score in analyses to synthetically balance the sample Stratification Regression Matching Weighting AcademyHealth, Seattle WA

14. June 25, 2006 Step 1: Estimate propensity scores Use logistic regression (or other method, e.g., CART) to estimate P(Z=1|X) = i, propensity score logit(Z) =X Focus is on prediction rather than estimation. –Include all potential confounders, but leave out factors related only to the exposure or outcome (Brookhart et al, 2006, AJE) –Include interaction effects as needed –ROC curve can be used to evaluate fit, but doesnt provide insight about appropriate covariates the estimated propensity score for the i th individual AcademyHealth, Seattle WA

15. June 25, 2006 Step 1: Estimate propensity for depression proc logistic descending; model major=age male smoke obese somecoll sedentary cardio outofcontrol treatint rxrisk2 /outroc=roc; run; Estimated AUC=0.72 Propensity score missing for 6.6% AcademyHealth, Seattle WA

16. June 25, 2006 Step 1: Estimate propensity for depression proc logistic descending; model major=age male smoke obese somecoll sedentary cardio outofcontrol treatint rxrisk2 + missing value indicators /outroc=roc; run; Estimated AUC=0.72 None missing propensity score AcademyHealth, Seattle WA

17. June 25, 2006 Propensity Strata AcademyHealth, Seattle WA Strata 1 Strata 2 Strata 3 Strata 4 Strata 5 Strata 6 Strata 7 Not Depressed 826 22% 794 21% 775 21% 736 20% 339 9% 146 4% 133 4% 3749 88% Depressed 27 5% 58 11% 78 15% 116 23% 87 17% 67 13% 80 16% 513 12% Total853 20% 852 20% 853 20% 852 20% 426 10% 213 5% 4262

18. June 25, 2006 Step 2: Check covariate balance StrataNot depressedDepressedN all27.144.34262 1 4.87.4853 216.212.1852 323.126.9853 439.441.4852 551.651.7426 666.467.2213 778.273.5213 Percent Sedentary

19. June 25, 2006 Step 3: Incorporate Propensity Scores into Proportional Hazards Model 1.Regression: Proportional hazards across different levels of the propensity score 2.Stratification: Allow different baseline hazards across propensity strata 3.Matching: Allow different baseline hazards for each matched pair 4.Weighting: Assume a common baseline hazard, AcademyHealth, Seattle WA

20. June 25, 2006 Regression-adjustment in the PH model Assume proportionality: check this assumption using Shoenfeld residuals. AcademyHealth, Seattle WA

21. June 25, 2006 Schoenfeld Residuals Little evidence for non-proportional hazards in propensity scores. Correlation between Schoenfeld-residual and rank-time Depression: 0.02 Propensity: -0.06

22. June 25, 2006 PH Model Results AcademyHealth, Seattle WA MethodEstimateSe(estimate)HRP-value Min Adj0.770.142.16<0.001 Full Adj0.260.161.300.09 Regression0.250.141.280.08

23. June 25, 2006 Stratification-adjustment in the PH model Stratified likelihood j : censoring indicator (1 if death obs) mth strata AcademyHealth, Seattle WA

24. June 25, 2006 PH Model Results AcademyHealth, Seattle WA MethodEstimateSe(estimate)HRP-value Min Adj0.770.142.16<0.001 Fully Adj0.260.161.300.09 Regression0.250.141.280.08 Stratified0.240.141.270.10

25. June 25, 2006 Matched Propensity Score Analysis 1.Use the full sample to estimate propensity scores 2.Identify matched pairs based on linear predictor from the propensity model. Matching within ±0.25*SE(X ) is recommended by Rosenbaum & Rubin (1983, 1985) 3.Assess matching: differences between matched and unmatched individuals; balance within matched sample. 4.Analyze data, accounting for matching. AcademyHealth, Seattle WA

26. June 25, 2006 Matching-adjustment in the PH model only 2/513 depressed excluded Within each matched pair, only the first death contributes to the likelihood leading to additional loss of information. mth pair AcademyHealth, Seattle WA

27. June 25, 2006 PH model results AcademyHealth, Seattle WA MethodEstimateSe(estimate)HRP-value Min Adj0.770.142.16<0.001 Full Adj0.260.161.300.09 Regression0.250.141.280.08 Stratified0.240.141.270.10 Matching0.260.211.300.21

28. June 25, 2006 AcademyHealth, Seattle WA Weighting-adjustment in the PH model (IPW) Weighted partial Likelihood Function up-weight individuals with unexpected exposure Limits options for handling ties Performs best when weights are estimated (Qi, Wang, Prentice, JASA,2005)

29. June 25, 2006 PH model results AcademyHealth, Seattle WA MethodEstimateSe(estimate)HRP-value Unadjusted0.340.141.40<0.02 Min Adj0.770.142.16<0.001 Full Adj0.260.161.300.09 Regression0.250.141.280.08 Stratified0.240.141.270.10 Matching0.260.211.300.21 IPW0.360.091.43<0.005

30. June 25, 2006 Z X Y Covariate models Estimate the effect of Z on Y conditional on X

31. June 25, 2006 IPW does not depend on estimating effects of Y | (Z and X) Z X Y Covariate models Propensity Syntheticall y balances X across Z Propensity models: P(Z|X)

32. June 25, 2006 Combined Adjustments Regression adjust and weight. AcademyHealth, Seattle WA

33. June 25, 2006 PH Model Results AcademyHealth, Seattle WA MethodEstimateSe(estimate)HRP-value Min Adj0.770.142.16<0.001 Full Adj0.260.161.300.09 Regression0.250.141.280.08 Stratified0.240.141.270.10 Matching0.260.211.300.21 IPW0.360.091.43<0.005 IPW+Reg0.360.091.43<0.005

34. June 25, 2006 Doubly Robust Regression Model True Propensity Model True NoYes No Yes An approach that is robust to misspecification of the regression model OR the propensity model. AcademyHealth, Seattle WA

35. June 25, 2006 Doubly Robust Estimators Idea: weighted estimators use only observed outcomes. DR estimators incorporate unobserved outcomes through their expected values. Increase efficiency, increase robustness Adjusted Score Function: indicates observing the assigned (patient selected) treatment weighted score

36. June 25, 2006 Score Adjustment, i i is an augmentation term that is a function of the regression model, M(Y|X, ) where Y=(, T):

37. June 25, 2006 Doubly Robust Estimator Expected value is 0 if regression model is true Expected value is 0 if propensity model is true

38. June 25, 2006 Doubly Robust Estimates Can calculate DR estimates iteratively: 1.Calculate starting values using PH 2.Estimate i via simulation given M(Y|X, ) and current parameter estimates, including baseline hazard (e.g., Nelson-Aalen estimators) 1.Use Newton-Raphson to solve the adjusted score for + TS approx

39. June 25, 2006 PH model results MethodEstimateSe(estimate)HRP-value Min Adj0.770.142.16<0.001 Full Adj0.260.161.300.09 Regression0.260.161.300.09 Stratified0.240.141.270.10 Matching0.260.211.300.21 IPW0.360.091.43<0.005 DR AcademyHealth, Seattle WA

40. June 25, 2006 Propensity Adjustment Compared to Inclusion of Covariates Separate models for treatment assignment and outcome. Focus on synthetic balance of sample. Maintain power while adjusting for many covariates –Need about 10-15 events per independent variable examined Multiple ways to adjust, allowing different assumptions about proportionality of hazards Can no longer make inference about individual covariates

41. June 25, 2006 Propensity Adjustment for Survival Models Omitting covariates from PH models may result in attenuation of estimates for included covariates (Mitra & Heitjen, Stat in Med, 2006). Covariate adjustment in PH model may reduce bias in estimates of covariate effects ( Lagakos & Shoenfeld, Biometrics, 1984 ) but has little effect on the variance of estimates. ( Anderson & Flemming, Biometrika, 1995 )

42. June 25, 2006 Propensity Adjustment for Survival Models: Recent Work Sturmer et al. AJE, 2005, Develop a regression- calibration approach to adjust for error in estimated propensity scores. Mitra & Heitjen, Stat in Med, 2006, develop a method for determining the effect an umeasured confounder would need to have to explain observed differences.

43. June 25, 2006 Propensity Models Additional research: More than two treatment/exposure groups Leon AC, Mueller TI, Solomon DA, Keller MB. 2001, Stat Med. Luellen JK, Shadish WR, & Clark MH. 2005, Evaluation Review, & references therein Imbens G. Biometrika, 2000. Continuous treatment/exposure measures AcademyHealth, Seattle WA